The system (REF ) reduce to the CNS equations by taking \(\psi \equiv 0\) . In order to study about the (REF ), we cite some reference about the CNS equations. The first local existence and uniqueness results were obtained by J. Nash [1]} for smooth initial data without vacuum. Later on, A. Matsumura and T. Nishida [2]} proved the global well-posedness and the time decay rate for smooth data close to equilibrium for \(d=3\) . In [3]}, A. V. Kazhikhov and V. V. Shelukhin established the first global existence result with large data in one dimensional space under some suitable condition on \(\mu \)  and \(\lambda \) . If \(\mu \)  is constant and \(\lambda (\rho )=b\rho ^\beta \) , X. Huang and J. Li[4]} obtained a global existence and uniqueness result for large initial data in two dimensional space(See also [5]}). In [6]}, X. Huang, J. Li, and Z. Xin proved the global well-posedness with vacuum. The blow-up phenomenons were studied by Z. Xin et al
in [7]}, [8]}, [9]}. Concerning the global existence of weak solutions for the large initial data, we may refer to [10]}, [11]}, [12]}, [13]}.
